Coefficient growth in square chains

Abstract

Suppose $((\cdots ((x^2-c_1{)}^2-c_2{)}^2\cdots {)}^2-c_{k-1}{)}^2-c_k$ splits into linear factors over $\mathbb{Z}$ and $c_k\ne 0$. We show that for each $j$ and each prime $p$, if $p\le 2^{j-1}$ then $p$ divides $c_j$. Consequently, $$\ln c_j>\frac{1}{4}\cdot 2^j\mathrm{for}j\ge 5$$ If we also have $p\equiv 3(\mathrm{mod}4)$ then $p^{2^{j-\lceil \lg p\rceil }}$ divides $c_j$. Consequently, if $k\ge 3$, there exists some absolute constant $\hat{I}»>0$ so that, $$\ln c_j>\hat{I}»k{2}^j\mathrm{for}\mathrm{all}j$$ These estimates argue against the possibility of explicitly constructing polynomials of the given form for large $k$, as the coefficients quickly become too large to manipulate.